# HW 7 - MATH 225 HOMEWORK 7 pp 291 293#1 3 6 12 14 16 23 26...

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MATH 225 HOMEWORK 7 pp. 291 – 293 #1, 3, 6, 12, 14, 16, 23, 26 – 28, 30, 31, 37 pp. 300 – 301 #1 – 4, 9, 12, 15, 16 – 21, 24 – 26, 28; 32, 39 p. 306 #3, 5 A. In M n ( F ) show that W = { A = [ a ij ] | for each i , a i 1 + a i 2 + + a in = 0} is a subspace of M n ( F ), find a basis for W , and determine dim F W . B. If A M n ( F ) show that B = {row 1 A , row 2 A , . . . , row n A } is a basis for F 1 xn if and only if C = {col 1 A , col 2 A , . . . . , col n A } is a basis for F n . C. Recall that P n = { a n x n + + a 1 x + a 0 R [ x ]}. Let W = { p ( x ) P 5 | p (1) = 0, p' (–1) = 0, and p" (1) = 0}. Show that W is a subspace of P 5 and find its dimension and a basis for it. D. Let A = { α 1 , . . . , α n }, B = { β 1 , . . . , β n }, and C = { γ 1 , . . . , γ n } be ordered bases of V F . Show that [ I ] AC = [ I ] AB [ I ] BC as follows: use ε j = [ γ j ] C , col j [ I ] AC = [ γ j ] A , and that [ γ j ] A = [ I ] AB ([ I ] BC [ γ j ] C ). Recall that for A M n ( R ), the image of A is Im( A ) = { B R n | AX = B has a solution} = colspace( A ), and the null space, or kernel of A , ker( A ) = N( A ) = { C R n | AC = 0 nx 1 }––the solutions of the homogeneous system AX = 0 R n . Each of these two is a subspace of R n . E. If A = " 1 " 2 " 1 " 1 1 1 0 0 0 2 0 1 0 " 2 2 0 # \$ % % % & ' ( ( ( M 4 ( R ), find conditions for vectors to be in each of the subspaces Im( A ) and N(
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